One thing that might stand out is a complete lack of interest in doing anything besides studying. It seems like it may be a sign of low social intelligence

Why is that necessary to get educated?

It's not.

But elite schools select for (1) students who will enrich the experience for their classmates and (2) who are likely to have an outsized impact on the world as adults and bring prestige to the institution.

There are some people who are so quantitatively intelligent that a lack of social skills/intelligence doesn't matter (geniuses), but these people are so exceptional they can't be really measured by a test like the SAT. It's too easy to get a perfect score.

Instead they'll distinguish themselves by excelling at STEM competitions (eg. Math Olympiad, programming contests, the Putnam).

Putnam is a collegiate event, on which most students in the math department of elite colleges (except MIT) score approximately 0.